Values of Graph-Restricted Games

1986 ◽  
Vol 7 (2) ◽  
pp. 210-220 ◽  
Author(s):  
Guillermo Owen
Keyword(s):  
Restricted Games

Enumerating Connected Subgraphs and Computing the Myerson and Shapley Values in Graph-Restricted Games

2019 ◽  
Vol 10 (2) ◽  
pp. 1-25 ◽  
Author(s):  
Oskar Skibski ◽  
Talal Rahwan ◽  
Tomasz P. Michalak ◽  
Michael Wooldridge
Keyword(s):  
Shapley Values ◽  
Connected Subgraphs ◽  
Restricted Games

Complexity of inheritance of ℱ-convexity for restricted games induced by minimum partitions

2020 ◽  
Vol 54 (1) ◽  
pp. 143-161
Author(s):  
A. Skoda
Keyword(s):  
Cooperative Game ◽  
Polynomial Time ◽  
Minimum Weight ◽  
Connected Components ◽  
Communication Graph ◽  
Convex Game ◽  
Restricted Game ◽  
Restricted Games ◽  
Unweighted Graph

Let G = (N, E, w) be a weighted communication graph. For any subset A ⊆ N, we delete all minimum-weight edges in the subgraph induced by A. The connected components of the resultant subgraph constitute the partition 𝒫min(A) of A. Then, for every cooperative game (N, v), the 𝒫min-restricted game (N, v̅) is defined by v̅(A)=∑F∈𝒫min(A)v(F) for all A ⊆ N. We prove that we can decide in polynomial time if there is inheritance of ℱ-convexity, i.e., if for every ℱ-convex game the 𝒫min-restricted game is ℱ-convex, where ℱ-convexity is obtained by restricting convexity to connected subsets. This implies that we can also decide in polynomial time for any unweighted graph if there is inheritance of convexity for Myerson’s graph-restricted game.

scholarly journals Convexity of graph-restricted games induced by minimum partitions

2019 ◽  
Vol 53 (3) ◽  
pp. 841-866 ◽  
Author(s):  
Alexandre Skoda
Keyword(s):  
Necessary Conditions ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Connected Components ◽  
Weighted Graphs ◽  
Classical Definition ◽  
Restricted Game ◽  
Definition Of ◽  
Adjacent Cycles ◽  
Restricted Games

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components corresponding to a minimum partition. This minimum partition 𝒫min is i nduced by the deletion of the minimum weight edges. We provide five necessary conditions on the graph edge-weights to have inheritance of convexity from the underlying game to the restricted game associated with 𝒫min. Then, we establish that these conditions are also sufficient for a weaker condition, called ℱ-convexity, obtained by restriction of convexity to connected subsets. Moreover, we prove that inheritance of convexity for Myerson restricted game associated with a given graph G is equivalent to inheritance of ℱ-convexity for the 𝒫min-restricted game associated with a particular weighted graph G′ built from G by adding a dominating vertex, and with only two different edge-weights. Then, we prove that G is cycle-complete if and only if a specific condition on adjacent cycles is satisfied on G′.

ON THE COMPLEXITY OF COMPUTING VALUES OF RESTRICTED GAMES

2002 ◽  
Vol 13 (05) ◽  
pp. 633-651 ◽  
Author(s):  
J. M. BILBAO ◽  
J. R. FERNÁNDEZ ◽  
J. J. LÓPEZ
Keyword(s):  
Cooperative Games ◽  
Partially Ordered Set ◽  
Polynomial Algorithm ◽  
Ordered Set ◽  
Combinatorial Structure ◽  
Distributive Lattices ◽  
Communication Graph ◽  
Partially Ordered ◽  
Connected Subgraphs ◽  
Restricted Games

The aim of this paper is to compute Shapley's and Banzhaf's values of cooperative games restricted by a combinatorial structure. There have been previous models developed to study the problem of games with partial cooperation. Games restricted by a communication graph were introduced by Myerson and Owen. Another type of combinatorial structure introduced by Gilles, Owen and van den Brink is equivalent to a subclass of antimatroids. Cooperative games in which the set of players is a partially ordered set, that is, games on distributive lattices was investigated by Faigle and Kern. We introduce a new combinatorial structure called augmenting system which is a generaligation of the antimatroid structure and the system of connected subgraphs of graph. We present new algorithmic procedures for computing values of games under augmenting systems restrictions and we show that there exist problems with polynomial algorithm complexity.

Closure spaces and restricted games

1998 ◽  
Vol 48 (1) ◽  
pp. 57-69
Author(s):  
Jesús Mario Bilbao
Keyword(s):  
Closure Spaces ◽  
Restricted Games

Communication and its cost in graph-restricted games

1988 ◽  
Vol 25 (3) ◽  
pp. 275-286 ◽  
Author(s):  
Edward C. Rosenthal
Keyword(s):  
Restricted Games
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